Different Types of Sets on the Basis of Number of Elements – Class 11 Applied Mathematics

Empty Set (Null Set)

An empty set is a set that contains no elements at all. It is denoted by the symbol ∅ or {}. It is important to note that {0} is not an empty set because it contains one element, zero.

n(∅) = 0

Example: Solved Example: Identify the empty set

Show Step-by-Step Solution

Step 1: Consider the set A = {x : x² = 4 and x is odd}.
Step 2: Solve x² = 4 to get x = 2 or x = -2.
Step 3: Check if either 2 or -2 is odd. Neither is odd.
Answer: A = ∅, hence it is an empty set.

Finite Set

A finite set is a set where the process of counting elements comes to an end. The number of elements in a finite set is called its cardinal number, denoted by n(A).

n(A) = k, where k ∈ {0, 1, 2, ...}

Example: Solved Example: Determine the cardinality

Show Step-by-Step Solution

Step 1: Let set B = {x : x is a prime number less than 10}.
Step 2: List the elements: B = {2, 3, 5, 7}.
Step 3: Count the elements: n(B) = 4.
Answer: B is a finite set with n(B) = 4.

Infinite Set

An infinite set is a set that is not finite. The number of elements in an infinite set cannot be expressed as a natural number, as the counting process never terminates.

n(A) = ∞

Example: Solved Example: Verify if a set is infinite

Show Step-by-Step Solution

Step 1: Consider the set C = {x : x is a multiple of 5}.
Step 2: List the elements: C = {5, 10, 15, 20, ...}.
Step 3: Observe that the list continues indefinitely.
Answer: C is an infinite set.

Section A: Multiple Choice Questions (MCQs)

easyIf A = {x : x is a month of the year starting with the letter J}, find n(A).

A. 1 B. 2 C. 3 D. 4

Show Answer

✔ Correct: C

The months are January, June, and July. Thus, A = {January, June, July}, so n(A) = 3.

easyWhich of the following sets is an empty set?

A. {x : x² = 1, x is even} B. {x : x² = 4, x is even} C. {0} D. {x : x is a prime number}

Show Answer

✔ Correct: A

x² = 1 implies x = 1 or x = -1. Neither is even. Thus, the set is empty.

mediumIf A = {x : x ∈ N, x < 5} and B = {x : x ∈ N, x is a prime number < 5}, find n(A) + n(B).

A. 4 B. 6 C. 7 D. 8

Show Answer

✔ Correct: B

A = {1, 2, 3, 4}, n(A) = 4. B = {2, 3}, n(B) = 2. Sum = 4 + 2 = 6.

mediumLet A = {x : x is a factor of 12}. Find n(A).

A. 4 B. 5 C. 6 D. 7

Show Answer

✔ Correct: C

Factors of 12 are {1, 2, 3, 4, 6, 12}. There are 6 elements.

hardIf A = {x : x = 2ⁿ, n ∈ {1, 2, 3, 4}}, find n(A).

A. 3 B. 4 C. 5 D. 16

Show Answer

✔ Correct: B

Elements are 2¹, 2², 2³, 2⁴, which are {2, 4, 8, 16}. There are 4 distinct elements.

hardIf A = {x : x is a root of x² - 5x + 6 = 0}, find n(A).

A. 0 B. 1 C. 2 D. 3

Show Answer

✔ Correct: C

Solving x² - 5x + 6 = 0 gives (x-2)(x-3) = 0, so x = 2, 3. n(A) = 2.

very hardLet A = {x : x ∈ Z, x² ≤ 4}. Find n(A).

A. 3 B. 4 C. 5 D. 6

Show Answer

✔ Correct: C

Integers satisfying x² ≤ 4 are {-2, -1, 0, 1, 2}. There are 5 elements.

Section B: Assertion–Reason Questions

Assertion (A): The set A = {x : x is a real number and x² + 1 = 0} is an empty set.
Reason (R): There is no real number whose square is -1.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Since x² = -1 has no real solutions, the set contains no elements, making it an empty set.

Assertion (A): The set of all points on a line segment is a finite set.
Reason (R): A line segment contains an infinite number of points between any two distinct points.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: D

Assertion is false because a line segment contains infinitely many points, making it an infinite set.

Section C: Short Answer Questions

mediumGiven set A = {x : x is a letter in the word MATHEMATICS}, find: (i) The roster form of A (ii) n(A) (iii) Is A a finite set?

Show Answer & Solution

(i) A = {M, A, T, H, E, I, C, S}
(ii) n(A) = 8
(iii) Yes, because the number of elements is countable.

mediumGiven set B = {x : x ∈ N, x is a multiple of 3}, find: (i) First 4 elements (ii) Is B finite or infinite? (iii) n(B) if we restrict x < 15.

Show Answer & Solution

(i) {3, 6, 9, 12}
(ii) Infinite
(iii) B = {3, 6, 9, 12}, so n(B) = 4.

hardGiven set C = {x : x² - 9 = 0}, D = {x : x ∈ N, x² = 9}, E = {x : x ∈ Z, x² = 9}. Find: (i) n(C) (ii) n(D) (iii) n(E).

Show Answer & Solution

(i) C = {-3, 3}, n(C) = 2
(ii) D = {3}, n(D) = 1
(iii) E = {-3, 3}, n(E) = 2

Section D: Long Answer Questions

hardClassify the following sets as finite or infinite and justify: (i) A = {x : x ∈ N, x is prime} (ii) B = {x : x ∈ N, x < 1000} (iii) C = {x : x is a point on a circle} (iv) D = {x : x is a day of the week} (v) E = {x : x ∈ N, x > 500}.

Show Answer & Solution

Step 1: A = {2, 3, 5, 7, 11, ...} is infinite because primes never end.
Step 2: B = {1, 2, ..., 999} is finite as it has 999 elements.
Step 3: C is infinite because a circle consists of infinitely many points.
Step 4: D = {Mon, Tue, Wed, Thu, Fri, Sat, Sun} is finite with n(D) = 7.
Step 5: E = {501, 502, ...} is infinite as it continues without bound.

very hardLet A = {x : x = n/(n+1), n ∈ {1, 2, 3, 4, 5}}. (i) List elements of A. (ii) Find n(A). (iii) If we define B = {x : x = 1 - 1/(n+1), n ∈ {1, 2, 3, 4, 5}}, compare A and B. (iv) Find n(A ∪ B). (v) Is A ∪ B finite?

Show Answer & Solution

Step 1: A = {1/2, 2/3, 3/4, 4/5, 5/6}.
Step 2: n(A) = 5.
Step 3: B = {1-1/2, 1-1/3, 1-1/4, 1-1/5, 1-1/6} = {1/2, 2/3, 3/4, 4/5, 5/6}. A = B.
Step 4: A ∪ B = {1/2, 2/3, 3/4, 4/5, 5/6}, so n(A ∪ B) = 5.
Step 5: Yes, A ∪ B is finite because it contains exactly 5 elements.

Section E: Case-Based Question

★

Case Study: A library tracks the number of books borrowed by 5 students in a week: Student A: 2, Student B: 5, Student C: 0, Student D: 3, Student E: 0. Let S be the set of books borrowed by these students.

Sub-question 1: (i) Represent the set S of books borrowed by the students. (ii) Find n(S).

Show Solution

(i) S = {0, 2, 3, 5} (ii) n(S) = 4.

Sub-question 2: If a new student F borrows 5 books, update the set S and find the new cardinality.

Show Solution

New S = {0, 2, 3, 5}. Since 5 is already in the set, the set remains {0, 2, 3, 5}. n(S) = 4.

CBSE Board & Sample Paper Past Year Questions

📋 CBSE 2023 📝 1 Mark

If A = {x : x is a letter in the word 'SCHOOL'}, find n(A).

Show Answer & Solution

The letters are S, C, H, O, L. Thus A = {S, C, H, O, L}. n(A) = 5.

📋 CBSE 2022 📝 1 Mark

Which of the following is an empty set? A = {x : x² = 9, x is even}, B = {x : x is a prime number}.

Show Answer & Solution

✔ Answer: A

x² = 9 implies x = 3 or -3. Neither is even. Thus A is empty.

📋 CBSE Sample Paper 2024 📝 1 Mark

If A = {x : x ∈ N, x < 1}, is A finite or infinite?

Show Answer & Solution

Since there are no natural numbers less than 1, A = ∅. An empty set is a finite set.

📋 CBSE 2020 📝 1 Mark

Find the number of elements in the set A = {x : x is an integer, -2 < x < 2}.

Show Answer & Solution

The integers are -1, 0, 1. Thus n(A) = 3.

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